Abstract
Functions that are holomorphic and Lipschitz in a smoothly bounded domain enjoy a gain in the order of Lipschitz regularity in the complex tangential directions near the boundary. We describe this gain explicitly in terms of the defining function near points of finite type in the boundary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
\(\displaystyle \left\| u\right\| _{\alpha } = \left\| u\right\| _\infty + \sup \limits _{\begin{array}{c} x,y\in \Omega \\ x\ne y \end{array}}\, \dfrac{\left|u(x) - u(y)\right|}{\left|x-y\right|^\alpha }.\)
References
Chang, D.-C.E., Krantz, S.G.: Holomorphic Lipschitz functions and application to the \(\overline{\partial }\)-problem. Colloq. Math. 62(2), 227–256 (1991)
Catlin, D.W.: Estimates of invariant metrics on pseudoconvex domains of dimension two. Math. Z. 200(3), 429–466 (1989)
D’Angelo, J.P.: Real hypersurfaces, orders of contact, and applications. Ann. Math. (2) 115(3), 615–637 (1982)
D’Angelo, J.P.: Several Complex Variables and the Geometry of Real Hypersurfaces, Studies in Advanced Mathematics. CRC Press, Boca Raton (1993)
Herbort, G.: Logarithmic growth of the Bergman kernel for weakly pseudoconvex domains in \({ C}^{3}\) of finite type. Manuscr. Math. 45(1), 69–76 (1983)
Krantz, S.G.: Boundary values and estimates for holomorphic functions of several complex variables. Duke Math. J. 47(1), 81–98 (1980)
Krantz, S.G.: Lipschitz spaces, smoothness of functions, and approximation theory. Expo. Math. 1(3), 193–260 (1983)
Krantz, G.S.: On a theorem of Stein. Trans. Am. Math. Soc. 320(2), 625–642 (1990)
McNeal, J.D., Stein, E.M.: Mapping properties of the Bergman projection on convex domains of finite type. Duke Math. J. 73(1), 177–199 (1994)
McNeal, J.D.: Boundary behavior of the Bergman kernel function in \({ C}^2\). Duke Math. J. 58(2), 499–512 (1989)
McNeal, J.D.: Estimates on the Bergman kernels of convex domains. Adv. Math. 109(1), 108–139 (1994)
Nagel, A., Rosay, J.P., Stein, E.M., Wainger, S.: Estimates for the Bergman and Szegő kernels in \({ C}^2\). Ann. Math. (2) 129(1), 113–149 (1989)
Nagel, A., Stein, E.M., Wainger, S.: Boundary behavior of functions holomorphic in domains of finite type, part 1. Proc Nat Acad Sci USA 78(11), 6596–6599 (1981)
Ravisankar, S.: Lipschitz properties of harmonic and holomorphic functions, Ph.D. dissertation, The Ohio State University (2011)
Ravisankar, S.: Transversally Lipschitz harmonic functions are Lipschitz. Complex Var. Elliptic Equ. 58(12), 1685–1700 (2013)
Rudin, W.: Holomorphic Lipschitz functions in balls. Comment. Math. Helv. 53(1), 143–147 (1978)
Stein, E.M.: Singular integrals and estimates for the Cauchy-Riemann equations. Bull. Am. Math. Soc. 79, 440–445 (1973)
Acknowledgments
I am deeply grateful to Jeffery McNeal for introducing me to this beautiful part of several complex variables, the many discussions regarding this project, and for his generosity and guidance over the years. I started working on this project as part of my Ph.D. thesis [14] under his guidance at The Ohio State University and the results reported here were obtained during my stay at the Institute for Advanced Study. I wish to thank the Institute for Advanced Study for their hospitality. I also wish to thank Jiří Lebl for insightful discussions regarding this project and Yunus Zeytuncu for his comments on a draft of this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by The James D. Wolfensohn Fund.
Rights and permissions
About this article
Cite this article
Ravisankar, S. Tangential Lipschitz gain for holomorphic functions on domains of finite type. Math. Ann. 364, 439–451 (2016). https://doi.org/10.1007/s00208-015-1216-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-015-1216-x